Properties

Label 20.5.d.a
Level 20
Weight 5
Character orbit 20.d
Self dual yes
Analytic conductor 2.067
Analytic rank 0
Dimension 1
CM discriminant -20
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 20.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(2.06739926168\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q - 4q^{2} + 2q^{3} + 16q^{4} + 25q^{5} - 8q^{6} + 82q^{7} - 64q^{8} - 77q^{9} + O(q^{10}) \) \( q - 4q^{2} + 2q^{3} + 16q^{4} + 25q^{5} - 8q^{6} + 82q^{7} - 64q^{8} - 77q^{9} - 100q^{10} + 32q^{12} - 328q^{14} + 50q^{15} + 256q^{16} + 308q^{18} + 400q^{20} + 164q^{21} - 878q^{23} - 128q^{24} + 625q^{25} - 316q^{27} + 1312q^{28} - 1198q^{29} - 200q^{30} - 1024q^{32} + 2050q^{35} - 1232q^{36} - 1600q^{40} + 482q^{41} - 656q^{42} - 2078q^{43} - 1925q^{45} + 3512q^{46} + 4402q^{47} + 512q^{48} + 4323q^{49} - 2500q^{50} + 1264q^{54} - 5248q^{56} + 4792q^{58} + 800q^{60} - 4078q^{61} - 6314q^{63} + 4096q^{64} - 4478q^{67} - 1756q^{69} - 8200q^{70} + 4928q^{72} + 1250q^{75} + 6400q^{80} + 5605q^{81} - 1928q^{82} + 8002q^{83} + 2624q^{84} + 8312q^{86} - 2396q^{87} + 4322q^{89} + 7700q^{90} - 14048q^{92} - 17608q^{94} - 2048q^{96} - 17292q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0
−4.00000 2.00000 16.0000 25.0000 −8.00000 82.0000 −64.0000 −77.0000 −100.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.5.d.a 1
3.b odd 2 1 180.5.f.b 1
4.b odd 2 1 20.5.d.b yes 1
5.b even 2 1 20.5.d.b yes 1
5.c odd 4 2 100.5.b.b 2
8.b even 2 1 320.5.h.a 1
8.d odd 2 1 320.5.h.b 1
12.b even 2 1 180.5.f.a 1
15.d odd 2 1 180.5.f.a 1
20.d odd 2 1 CM 20.5.d.a 1
20.e even 4 2 100.5.b.b 2
40.e odd 2 1 320.5.h.a 1
40.f even 2 1 320.5.h.b 1
60.h even 2 1 180.5.f.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.5.d.a 1 1.a even 1 1 trivial
20.5.d.a 1 20.d odd 2 1 CM
20.5.d.b yes 1 4.b odd 2 1
20.5.d.b yes 1 5.b even 2 1
100.5.b.b 2 5.c odd 4 2
100.5.b.b 2 20.e even 4 2
180.5.f.a 1 12.b even 2 1
180.5.f.a 1 15.d odd 2 1
180.5.f.b 1 3.b odd 2 1
180.5.f.b 1 60.h even 2 1
320.5.h.a 1 8.b even 2 1
320.5.h.a 1 40.e odd 2 1
320.5.h.b 1 8.d odd 2 1
320.5.h.b 1 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 2 \) acting on \(S_{5}^{\mathrm{new}}(20, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T \)
$3$ \( 1 - 2 T + 81 T^{2} \)
$5$ \( 1 - 25 T \)
$7$ \( 1 - 82 T + 2401 T^{2} \)
$11$ \( ( 1 - 121 T )( 1 + 121 T ) \)
$13$ \( ( 1 - 169 T )( 1 + 169 T ) \)
$17$ \( ( 1 - 289 T )( 1 + 289 T ) \)
$19$ \( ( 1 - 361 T )( 1 + 361 T ) \)
$23$ \( 1 + 878 T + 279841 T^{2} \)
$29$ \( 1 + 1198 T + 707281 T^{2} \)
$31$ \( ( 1 - 961 T )( 1 + 961 T ) \)
$37$ \( ( 1 - 1369 T )( 1 + 1369 T ) \)
$41$ \( 1 - 482 T + 2825761 T^{2} \)
$43$ \( 1 + 2078 T + 3418801 T^{2} \)
$47$ \( 1 - 4402 T + 4879681 T^{2} \)
$53$ \( ( 1 - 2809 T )( 1 + 2809 T ) \)
$59$ \( ( 1 - 3481 T )( 1 + 3481 T ) \)
$61$ \( 1 + 4078 T + 13845841 T^{2} \)
$67$ \( 1 + 4478 T + 20151121 T^{2} \)
$71$ \( ( 1 - 5041 T )( 1 + 5041 T ) \)
$73$ \( ( 1 - 5329 T )( 1 + 5329 T ) \)
$79$ \( ( 1 - 6241 T )( 1 + 6241 T ) \)
$83$ \( 1 - 8002 T + 47458321 T^{2} \)
$89$ \( 1 - 4322 T + 62742241 T^{2} \)
$97$ \( ( 1 - 9409 T )( 1 + 9409 T ) \)
show more
show less